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- Consider the following three-player game (player 3 chooses A or B, the payoffs are listed in order of the player numbers): Player 2 Player 2 L R U 2, 2,0 5,5, 5 U 4, 4, 1 4, 2, 8 Player 1 Player 1 D 8, 6, 8 0,7,4 D 0, 2,9 4, 2, 5 A 1. Find the pure-strategy Nash equilibria of this game. 2. Consider the two-period repeated game (the repeated game payoffs are simply the sums of the payoffs in the two periods). Find the subgame-perfect equilibria of the repeated game and list the set of payoffs.Consider the following three-Player, extensive-form game - (-2,–1,2) а 2 b 3 Y +(-2,0,0) A (2,0,1) Y 3 (0,0,1) ►(0,0,0) a (0,1,0) (1,1,1) a) Find all pure strategy Nash equilibria of this game. b) Find all pure strategy SPNE of this game.D Question 11 Consider the following Extended Form Game: O 1 02 O 3 A' 04 (9,5) A P2 P1 B' A' (5,2) (8,5) B x A (1,6) P2 B' According to backwards induction, in how many turn does the game end? P1 B (6,7)
- Consider the Stackelberg game depicted below in which you are the row player. R U 4,0 1,2 3,2 0,1 0,0 2,0 You may choose whether you want to be the leader (and commit to a possibly mixed strategy) or the follower in a game against the course staff (column player). You may trust that we maximize our expected payoff. The points awarded to you will equal one half of the expected payoff you obtain. If you want to be the leader, please submit your commitment strategy. For example, if you want to commit to [0.5: U, 0.2: M, 0.3: D], then submit: 0.5 0.2 0.3 If you want to be the follower instead, just submit: FConsider the following three-player game represented in extensive form: Player 3 L Player 2 Player 1 R Player 2 U D Player 3 (0,2,0) U D U D (0, 1, 1) (0, 3, k) (0, 4, 3) (0,4,2) U D How many pure-strategy subgame-perfect Nash equilibria are there in this game if k > 1? 5 3 7 1 It depends on the value of k. (2,2, 0)2. Paul and Stella play a game with three strategies each, T, M, and B for Paul, and L, C, and R for Stella. Both move simultaneously. The payoffs are given by the following form: Stella L с R Paul T (8,4) (10, 2) (2,3) M (4,2) (10, 1) (5,7) B (1,4) (10, 3) (9,-4) a. Which strategy is dominated? b. What is the pure-strategy Nash equilibrium? Identify all if there are more than one. c. If Paul moved first, so that Stella observed it, which strategy would Paul choose?
- Consider the following game. Firm 1 can implement one of two actions, A or B. Firm 2 observes the action chosen by Firm 1 and then decides whether to fight it or not. (-10, 20) F2 Fight A Don't fight -(30, 10) Firm 1 (-10, 0) Fight B Don't fight -(20, 15) (a) Consider the following strategy profile: Firm 1 chooses A; Firm 2 chooses fight if A, and fight if B. • this strategy profile is [Select] (b) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses fight if A, and don't fight if B. • this strategy profile is [Select] O (c) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses don't fight if A and don't fight if B. • this strategy profile is [Select] 00 F2 ()3. Consider the game represented by the payoff matrix below: R S (12, 12) (8,-5) (25, 14) (50, 12) (26,70) (-2.6) (20, 100) (22,75) C (11,0) (3,25) (24, 26) (50,26) D (5,-10) (16,-8) (13,-8) (50,-20) A B a. Find the Nash equilibria of the game using any suitable method. b. Apply Iterated Elimination of Strictly Dominated Actions (IESDA) to this game. Explain the steps as necessary. Do the Nash equilibria found in part a. survive IESDA? c. Apply Iterated Elimination of (Weakly) Dominated Actions (IEDA) to this game by elim- inating all whakly dominated actions in each round. Explain the steps as necessary. Do the Nash equilibria found in part a, survive IEDA16. Boeing is considering whether it should develop its next aircraft with a wide body (WIDE) or whether it should make it a long aeroplane (LONG). Airbus, Boeing's competitor, also has a choice between choosing WIDE or LONG. The payoffs to the two companies are shown in the payoff matrix below. What are the (pure strategy) Nash equilibrium in this game? (The first action in the brackets is the strategy of Boeing, while the second is Airbus's strategy.) Airbus WIDE LONG WIDE 40 for Airbus Boeing 50 for Boeing O for Airbus 0 for Boeing LONG 0 for Airbus 0 for Boeing 50 for Airbus 40 for Boeing a. (WIDE, WIDE) for Boeing and Airbus respectively) and (WIDE, LONG) b. (LONG, WIDE) and (WIDE, LONG) c. (WIDE, WIDE) d. (LONG, LONG) e. (WIDE, WIDE) and (LONG, LONG)
- Consider the Normal Form Game characterized in the following figure: P1 \ P2 b1 b2 b3 a1 (1,2) (0,1) (0,3) a2 (3,-1) (1,1) (-1,0) a3 (2,2) (1,-1) (3,0) a4 (-5,1) (-1,0) (-3,-2) a5 (-3,3) (3,0) (2,4) What is the set of actions of player 1? O la 1, a2, a3, a4, a5) O (b1,b2, b3} O {a2, a3, a4) O (bl, b2}Consider the following NFG: X A (a,b) B (1/2a, 2b) (2a, b) Y (1/2a, 2b) Assume that a > 0,b>0. Which is the complete set of Nash Equilibria in mixed strategies? ((1,0), (1,0)),((- 4'4 ○ ((1,0), (1,0)), ((0,1),(0,1)) a b °(( 45² 45), (², 2)) a+b a+bConsider the extensive form game shown below. At each terminal node, the first payoff belongs to player 1, the second payoff belongs to player 2, and the third payoff belongs to player 3. How many strategies does player 3 have in this game? Numerical answer (12 (2,2,1) 2 b₂ (22, (2,4,2) (4,2,3) b₁ 2 (3,3,3) C1 02/ 03 b2 b3 (2,2,1) (4,2,0) (2,0,2) 03 3 b3 (0,3,4) image 1
